Fubar is a terminology used during World War II to describe a mission that had been severely messed up or an event was complete chaos.
In The Film
It was constantly used by Captian Miller and his men during their search for Ryan. A soldier called Lieutenant Dewindt knew what it meant as well.
Only Upham did not know what it meant until Mellish indirectly told him when they were having a conversation whilst collecting ammunition. This was also the last time "Fubar" was said in the film.
Meaning
There are two meanings. They are:
Fouled Up Beyond All Recognition (clean version).
F**ked Up Beyond All Recognition (what the army commonly meant it to be).